Questions C4 (1219 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 PURE S1 S2 S3 S4 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Sort by: Default | Easiest first | Hardest first
OCR C4 2008 January Q5
8 marks Standard +0.3
5 The vector equations of two lines are $$\mathbf { r } = ( 5 \mathbf { i } - 2 \mathbf { j } - 2 \mathbf { k } ) + s ( 3 \mathbf { i } - 4 \mathbf { j } + 2 \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = ( 2 \mathbf { i } - 2 \mathbf { j } + 7 \mathbf { k } ) + t ( 2 \mathbf { i } - \mathbf { j } - 5 \mathbf { k } ) .$$ Prove that the two lines are
  1. perpendicular,
  2. skew.
OCR C4 2008 January Q6
8 marks Standard +0.3
6
  1. Expand \(( 1 + a x ) ^ { - 4 }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).
  2. The coefficients of \(x\) and \(x ^ { 2 }\) in the expansion of \(( 1 + b x ) ( 1 + a x ) ^ { - 4 }\) are 1 and - 2 respectively. Given that \(a > 0\), find the values of \(a\) and \(b\).
OCR C4 2008 January Q8
8 marks Standard +0.3
8 Water flows out of a tank through a hole in the bottom and, at time \(t\) minutes, the depth of water in the tank is \(x\) metres. At any instant, the rate at which the depth of water in the tank is decreasing is proportional to the square root of the depth of water in the tank.
  1. Write down a differential equation which models this situation.
  2. When \(t = 0 , x = 2\); when \(t = 5 , x = 1\). Find \(t\) when \(x = 0.5\), giving your answer correct to 1 decimal place.
OCR C4 2008 January Q9
9 marks Standard +0.3
9 The parametric equations of a curve are \(x = t ^ { 3 } , y = t ^ { 2 }\).
  1. Show that the equation of the tangent at the point \(P\) where \(t = p\) is $$3 p y - 2 x = p ^ { 3 } .$$
  2. Given that this tangent passes through the point ( \(- 10,7\) ), find the coordinates of each of the three possible positions of \(P\).
OCR C4 2008 January Q10
11 marks Standard +0.3
10
  1. Use the substitution \(x = \sin \theta\) to find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { 1 } { \left( 1 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } } \mathrm {~d} x$$
  2. Find the exact value of $$\int _ { 1 } ^ { 3 } \frac { \ln x } { x ^ { 2 } } \mathrm {~d} x$$ 4
OCR C4 2007 June Q1
5 marks Moderate -0.3
1 The equation of a curve is \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \frac { 3 x + 1 } { ( x + 2 ) ( x - 3 ) }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Hence find \(\mathrm { f } ^ { \prime } ( x )\) and deduce that the gradient of the curve is negative at all points on the curve.
OCR C4 2007 June Q2
6 marks Standard +0.3
2 Find the exact value of \(\int _ { 0 } ^ { 1 } x ^ { 2 } \mathrm { e } ^ { x } \mathrm {~d} x\).
OCR C4 2007 June Q3
6 marks Moderate -0.3
3 Find the exact volume generated when the region enclosed between the \(x\)-axis and the portion of the curve \(y = \sin x\) between \(x = 0\) and \(x = \pi\) is rotated completely about the \(x\)-axis.
OCR C4 2007 June Q4
7 marks Moderate -0.3
4
  1. Expand \(( 2 + x ) ^ { - 2 }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), and state the set of values of \(x\) for which the expansion is valid.
  2. Hence find the coefficient of \(x ^ { 3 }\) in the expansion of \(\frac { 1 + x ^ { 2 } } { ( 2 + x ) ^ { 2 } }\).
OCR C4 2007 June Q5
9 marks Moderate -0.3
5 A curve \(C\) has parametric equations $$x = \cos t , \quad y = 3 + 2 \cos 2 t , \quad \text { where } 0 \leqslant t \leqslant \pi$$
  1. Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\) and hence show that the gradient at any point on \(C\) cannot exceed 8 .
  2. Show that all points on \(C\) satisfy the cartesian equation \(y = 4 x ^ { 2 } + 1\).
  3. Sketch the curve \(y = 4 x ^ { 2 } + 1\) and indicate on your sketch the part which represents \(C\).
OCR C4 2007 June Q6
8 marks Standard +0.3
6 The equation of a curve is \(x ^ { 2 } + 3 x y + 4 y ^ { 2 } = 58\). Find the equation of the normal at the point \(( 2,3 )\) on the curve, giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.
OCR C4 2007 June Q7
10 marks Moderate -0.3
7
  1. Find the quotient and the remainder when \(2 x ^ { 3 } + 3 x ^ { 2 } + 9 x + 12\) is divided by \(x ^ { 2 } + 4\).
  2. Hence express \(\frac { 2 x ^ { 3 } + 3 x ^ { 2 } + 9 x + 12 } { x ^ { 2 } + 4 }\) in the form \(A x + B + \frac { C x + D } { x ^ { 2 } + 4 }\), where the values of the constants \(A , B , C\) and \(D\) are to be stated.
  3. Use the result of part (ii) to find the exact value of \(\int _ { 1 } ^ { 3 } \frac { 2 x ^ { 3 } + 3 x ^ { 2 } + 9 x + 12 } { x ^ { 2 } + 4 } \mathrm {~d} x\).
OCR C4 2007 June Q8
10 marks Moderate -0.3
8 The height, \(h\) metres, of a shrub \(t\) years after planting is given by the differential equation $$\frac { \mathrm { d } h } { \mathrm {~d} t } = \frac { 6 - h } { 20 }$$ A shrub is planted when its height is 1 m .
  1. Show by integration that \(t = 20 \ln \left( \frac { 5 } { 6 - h } \right)\).
  2. How long after planting will the shrub reach a height of 2 m ?
  3. Find the height of the shrub 10 years after planting.
  4. State the maximum possible height of the shrub.
OCR C4 2007 June Q9
11 marks Standard +0.3
9 Lines \(L _ { 1 } , L _ { 2 }\) and \(L _ { 3 }\) have vector equations $$\begin{aligned} & L _ { 1 } : \mathbf { r } = ( 5 \mathbf { i } - \mathbf { j } - 2 \mathbf { k } ) + s ( - 6 \mathbf { i } + 8 \mathbf { j } - 2 \mathbf { k } ) , \\ & L _ { 2 } : \mathbf { r } = ( 3 \mathbf { i } - 8 \mathbf { j } ) + t ( \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k } ) , \\ & L _ { 3 } : \mathbf { r } = ( 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k } ) + u ( 3 \mathbf { i } + c \mathbf { j } + \mathbf { k } ) . \end{aligned}$$
  1. Calculate the acute angle between \(L _ { 1 }\) and \(L _ { 2 }\).
  2. Given that \(L _ { 1 }\) and \(L _ { 3 }\) are parallel, find the value of \(c\).
  3. Given instead that \(L _ { 2 }\) and \(L _ { 3 }\) intersect, find the value of \(c\). 4
OCR C4 2008 June Q1
6 marks Moderate -0.3
1
  1. Simplify \(\frac { \left( 2 x ^ { 2 } - 7 x - 4 \right) ( x + 1 ) } { \left( 3 x ^ { 2 } + x - 2 \right) ( x - 4 ) }\).
  2. Find the quotient and remainder when \(x ^ { 3 } + 2 x ^ { 2 } - 6 x - 5\) is divided by \(x ^ { 2 } + 4 x + 1\).
OCR C4 2008 June Q2
5 marks Standard +0.3
2 Find the exact value of \(\int _ { 1 } ^ { \mathrm { e } } x ^ { 4 } \ln x \mathrm {~d} x\).
OCR C4 2008 June Q3
8 marks Standard +0.3
3 The equation of a curve is \(x ^ { 2 } y - x y ^ { 2 } = 2\).
  1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y ^ { 2 } - 2 x y } { x ^ { 2 } - 2 x y }\).
  2. (a) Show that, if \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\), then \(y = 2 x\).
    (b) Hence find the coordinates of the point on the curve where the tangent is parallel to the \(x\)-axis.
OCR C4 2008 June Q4
7 marks Standard +0.3
4 Relative to an origin \(O\), the points \(A\) and \(B\) have position vectors \(3 \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k }\) and \(\mathbf { i } + 3 \mathbf { j } + 4 \mathbf { k }\) respectively.
  1. Find a vector equation of the line passing through \(A\) and \(B\).
  2. Find the position vector of the point \(P\) on \(A B\) such that \(O P\) is perpendicular to \(A B\).
OCR C4 2008 June Q5
8 marks Standard +0.3
5
  1. Show that \(\sqrt { \frac { 1 - x } { 1 + x } } \approx 1 - x + \frac { 1 } { 2 } x ^ { 2 }\), for \(| x | < 1\).
  2. By taking \(x = \frac { 2 } { 7 }\), show that \(\sqrt { 5 } \approx \frac { 111 } { 49 }\).
OCR C4 2008 June Q6
8 marks Standard +0.3
6 Two lines have equations $$\mathbf { r } = \left( \begin{array} { r } 1 \\ 0 \\ - 5 \end{array} \right) + t \left( \begin{array} { l } 2 \\ 3 \\ 4 \end{array} \right) \quad \text { and } \quad \mathbf { r } = \left( \begin{array} { r } 12 \\ 0 \\ 5 \end{array} \right) + s \left( \begin{array} { r } 1 \\ - 4 \\ - 2 \end{array} \right) .$$
  1. Show that the lines intersect.
  2. Find the angle between the lines.
OCR C4 2008 June Q8
11 marks Standard +0.3
8
  1. Given that \(\frac { 2 t } { ( t + 1 ) ^ { 2 } }\) can be expressed in the form \(\frac { A } { t + 1 } + \frac { B } { ( t + 1 ) ^ { 2 } }\), find the values of the constants \(A\) and \(B\).
  2. Show that the substitution \(t = \sqrt { 2 x - 1 }\) transforms \(\int \frac { 1 } { x + \sqrt { 2 x - 1 } } \mathrm {~d} x\) to \(\int \frac { 2 t } { ( t + 1 ) ^ { 2 } } \mathrm {~d} t\).
  3. Hence find the exact value of \(\int _ { 1 } ^ { 5 } \frac { 1 } { x + \sqrt { 2 x - 1 } } \mathrm {~d} x\).
OCR C4 2008 June Q12
Moderate -0.3
12
0
5 \end{array} \right) + s \left( \begin{array} { r } 1
- 4
- 2 \end{array} \right) .$$
  1. Show that the lines intersect.
  2. Find the angle between the lines.
  3. Show that, if \(y = \operatorname { cosec } x\), then \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) can be expressed as \(- \operatorname { cosec } x \cot x\).
  4. Solve the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = - \sin x \tan x \cot t$$ given that \(x = \frac { 1 } { 6 } \pi\) when \(t = \frac { 1 } { 2 } \pi\). 8
  5. Given that \(\frac { 2 t } { ( t + 1 ) ^ { 2 } }\) can be expressed in the form \(\frac { A } { t + 1 } + \frac { B } { ( t + 1 ) ^ { 2 } }\), find the values of the constants \(A\) and \(B\).
  6. Show that the substitution \(t = \sqrt { 2 x - 1 }\) transforms \(\int \frac { 1 } { x + \sqrt { 2 x - 1 } } \mathrm {~d} x\) to \(\int \frac { 2 t } { ( t + 1 ) ^ { 2 } } \mathrm {~d} t\).
  7. Hence find the exact value of \(\int _ { 1 } ^ { 5 } \frac { 1 } { x + \sqrt { 2 x - 1 } } \mathrm {~d} x\). 9 The parametric equations of a curve are $$x = 2 \theta + \sin 2 \theta , \quad y = 4 \sin \theta$$ and part of its graph is shown below. \includegraphics[max width=\textwidth, alt={}, center]{b8ba126f-c5fa-4828-9439-e5162a03ca5b-3_646_1150_1050_500}
  8. Find the value of \(\theta\) at \(A\) and the value of \(\theta\) at \(B\).
  9. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sec \theta\).
  10. At the point \(C\) on the curve, the gradient is 2 . Find the coordinates of \(C\), giving your answer in an exact form.
OCR C4 Specimen Q1
4 marks Moderate -0.8
1 Find the quotient and remainder when \(x ^ { 4 } + 1\) is divided by \(x ^ { 2 } + 1\).
OCR C4 Specimen Q2
5 marks Moderate -0.8
2
  1. Expand \(( 1 - 2 x ) ^ { - \frac { 1 } { 2 } }\) in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\).
  2. State the set of values for which the expansion in part (i) is valid.
OCR C4 Specimen Q3
5 marks Moderate -0.3
3 Find \(\int _ { 0 } ^ { 1 } x \mathrm { e } ^ { - 2 x } \mathrm {~d} x\), giving your answer in terms of e.